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Application of the Todd-Coxeter Algorithm in the Computation of Group Theory

DOI: 10.4236/alamt.2023.133003, PP. 37-52

Keywords: Todd-Coxeter Algorithm, Subgroup, Semi-Direct, Operating Group, Homomorphism

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Abstract:

In this article, we have described the Todd-Coxeter algorithm. Indeed, the Todd-Coxeter algorithm is a mathematical tool used in the field of group theory. It makes it possible to determine different possible presentations of a group, i.e. different ways of expressing its elements and operations. We have also applied this algorithm to a subgroup generated H by G; where we obtained a table of the subgroup, three tables of relators including: Table of the relator aaaa; Table of the relator abab; Table of the relator bbb and a multiplication table aa'bb'. Once the algorithm is complete, the unit of H in G is 6. We have explicitly obtained a homomorphism of G in the group of permutations of H/G which is isomorphic to G6; where we have noticed that it is injective: in fact, an element of the nucleus belongs to the intersection of the xH x 1 for xG , in particular, it belongs to H; on the other hand, the image of H in G6 is of order 4, so the nucleus is reduced to the neutral element.

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